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An Introduction to heavy-tailed and subexponential distributions

An Introduction to Heavy-Tailed and Subexponential Distributions (1st edition 2011, 2nd edition 2013)

S. Foss, D. Korshunov, and S. Zachary

NOVEL RESULTS

Chapter 2 contains substantial novel material on the characterisation and properties of long-tailed distributions, especially their convolutions, notably Theorems 2.28, 2.29, 2.36, 2.37, 2.39. It further contains novel insensitivity characterisations of long-tailed distributions: here Theorems 2.47, 2.48 and 2.49 are all new.

Similarly Chapter 3 contains considerable novel material on the characterisation and properties of subexponential distributions, their precise relation to long-tailed distributions, and their convolutions, notably Thms 3.14, 3.15 and their Corollaries 3.16 and 3.17. That part of Theorem 3.33 which relates to h-insensitivity is new, and the proof of the entire result is based on a new idea which very much simplifies the original of Embrechts and Goldie.

Novel results in Chapter 5 include Theorem 5.1 (an elementary proof of lower bound for the distribution of the maximum of a random walk, with no conditions on the increments other than that it have a negative mean), and new results on the finite time asymptotics for the maximum of a random walk: Theorem 5.3 under the stated conditions, and Theorems 5.4 and 5.4* (2nd ed). All the material of Section 5.13 (2nd ed) on how random walks with heavy-tailed increments attain large values is based on an entirely new approach from that of the original of Asmussen and Kluppelberg.